With more of the city’s already limited hospitals destroyed in the latest offensive, Dr. Richard Brennan, the organization’s director of emergency response, told reporters that many of the 846 people wounded were expected to die for lack of treatment. That includes 261 children, he said.

There were competing claims about the success of the government’s offensive on Friday, with the government saying its fighters had captured more territory in northern parts of the city and penetrated the center of Aleppo, where they were clashing with the rebels.

Rebel commanders denied the reports, however, saying government forces had not made any progress since capturing the Handarat refugee camp north of the city on Thursday.

International monitoring groups have accused Russia and Syria of bombing civilian targets, like hospitals, schools and basic infrastructure, without regard to the safety of the civilian population of 250,000 in eastern Aleppo. On Friday, the rebels said the government bombed the main water plant that supplies the area.

After five years of civil war, the Syrian Army has been reduced to just a few effective and experienced units, analysts say, and is no longer capable of prosecuting a broad offensive on its own. That has created a void that is being filled by groups from outside the country: Shiite militias and mercenaries from Iraq and Afghanistan, fighters with the Lebanese militant group Hezbollah, and soldiers from Iran’s Islamic Revolutionary Guards Corps and its elite Quds Force.

Those militias have been backed by a Russian air campaign that has “hit civilian areas and increasingly used indiscriminate weapons, including cluster and incendiary munitions,” Britain’s special representative to Syria, Gareth Bayley, said in a statement Friday. Others have accused Russia of using bunker buster bombs meant for military targets, while the American ambassador to the United Nations, Samantha Power, has called the attacks “barbarism.”

Russia has defended its actions as necessary to thwart terrorists, and its foreign minister, Sergey V. Lavrov, told the BBC on Friday that Russian forces were not using “any munition which is prohibited by the United Nations.”

Mr. Lavrov said that Russia was taking “all necessary precautions” not to hit civilians, and that “If this happens, then we are very sorry.”

Russia has also accused the United States, under the guise of backing the opposition in the civil war, of abetting terrorism and helping militant groups like the Nusra Front, an affiliate of Al Qaeda, and the Islamic State, also known as ISIS or ISIL.

Mr. Lavrov said that Washington had “pledged solemnly to take as a priority an obligation to separate the opposition from Nusra,” but that the United States had shown that it was “not able or not willing to do this.”

A frustrated Secretary of State John Kerry has grown increasingly dismissive of Russia’s denials and justifications, and this week threatened to end talks on Syria unless the bombing stopped.

Abu Anas, the nom de guerre of a former opposition fighter who joined the Nusra Front, which now calls itself the Levant Conquest Front, said Friday that the brutal attacks were actually helping their cause by radicalizing the population.

Friday was the first anniversary of Moscow’s intervention in the Syrian civil war, an occasion that prompted a number of reports on civilian casualties in Russian airstrikes. The Syrian Observatory for Human Rights, a monitoring group based in Britain, put the figure at 3,189, while the independent website Airwars has said that at least 3,000 have died.

Until a few days ago, eastern Aleppo had eight working hospitals, but its two biggest medical centers came under attack this week and were no longer functioning, said Dr. Brennan of the W.H.O.

Four children died from their wounds in the last few days, Dr. Brennan said, because of a shortage of space in intensive care units.

As of two days ago, the United Nations believed that there were 35 doctors struggling to keep up with the flow of casualties but Dr. Brennan said it now believed that there were only 30. “These guys are exhausted, they are drained physically and emotionally,” he said.

The United Nations has drawn up plans for evacuations and identified hospitals in northern Syria that could receive the wounded. But those plans have been held in abeyance because of the intense fighting, the absence of assurances of safe passage and the lack of a green light from the government.

A convoy of trucks carrying aid to besieged Syrians was attacked on Sept. 19, killing about 20 people, and preventing food, blankets and clothing from reaching people in rebel-held areas of Aleppo.

Ban Ki-moon, the United Nations secretary-general, announced Friday that he would establish a board of inquiry to investigate the attack on the convoy. United Nations officials have said the attack could be a war crime.

The United Nations has medicines and medical equipment in position and ready to move to besieged Syrians in east Aleppo, Dr. Brennan said Friday, but the aid cannot get into the city.

“We are asking for four things: stop the killing, stop attacks on health care, let the sick and wounded out, and let the aid in,” Dr. Brennan said, describing conditions in the city as “beyond unimaginable.”

“The message is simple,” he added. “We hope this time it doesn’t fall on deaf ears.”

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